Cochin University of Science and Techology %28CUST%29 2007 B.Tech Civil Engineering Engineering Mathematics II - Question Paper

IT/CS/EC/CE/ME/SE/EB/EI/EE 301 ENGINEERING MATHEMATICS H

(2006 Admissions)

Maximum Marks: 100

PART A

(Answer ALL questions)

(All questions carry FIVE marks)

(8x5 = 40)

Define rank of a matrix. Find the values of / and m .such that the rank of the matrix

(a)

2 1-13 1-12 4 7 -1 / m

Let T be a linear transformation from R³ into R defined by 7^T( , x₂, x₃) = x*, x₂², x². Show that T is not a linear transformation.

Obtain the half range sine series for e^x in 0 < x < 1.

₍ . [1 for \x <1    rsinx ,

Find the Fourier transform of t (X1 = <    . Hence evaluate - ax.

10 for x > 1    * x

s² +s

Find the inverse Laplace transform of log

U+⁴

Find the Laplace transform of the saw toothed wave of period T, given /(0 = p</<7\

If u x² + y² + z² and V -xi + yj + zk, show that div (wvj = 5m .

Find the work done when a force F = (x² y² + x) i (2xy + y) j moves a particle in the xy-plane from (0,0)to(l,l) along the parabola y² =x.

PART B

(All questions carry FIFTEEN marks)

-2 2 -3

2 1 -6

_w-l -2 0 to the largest eigen value.

Find ker(r) and ran(T} and their dimensions where T:R³ > R³ defined by

		rx + y
y	*	z
<z)		i H

OR

represented by A⁸ 5 A* +7 A⁶ - 3 A⁵ + A⁴ 5 A³ + 8 A² 2 A +1. (b) Test for consistency and solve the following

systemx - y + z = 1,2x + y - z = 2,5x - 2y + 2z = 5.

Find the Fourier sine transform of Hence show that

x sin mx n ___m _

I _ ox =e ,m > U. i \ + x² 2

OR

, si \ .    /    ri \ fOfor ~2<x<0

Expand / I x) in Fourier series in the interval I 2,2) when / (JC1

^W    '    ^W [1 for 0<x<2

, fl Oxctt Express the function J I    asa Fourier sine integral and hence

[0 X>7t

_f(l-COSl)

evaluate I--- sm x X a a .

j X .

Find the Laplace transforms _e~^a) _

(i) --(ii) sin/M(/-;r.)

Solve by the method of Laplace transforms y^m + 2 y* y' 2y = 0 given

(0) = /(0) = 0 and y(0) = 6.

t

Evaluate jt e~^2t s'mt dt. o

OR

Define a unit impulse function and find its Laplace transform.

r-l f    ¹    1

Apply Convolution theorem to evaluate L \    , _x ,.

l(^{J + fl})(^{J +} *)J

Find the Laplace transform of the periodic function of period 2a defined by ( . J 1 for 0t<a

11 for a<t<2a

Prove that jj = 0.

Show that F.dr = 3n₉ given that F =* zi + xj + yk and C being the arc of the

curve r = cos// + siiUj + tk from t = 0 to t = n.

If F = (lx² ~3zi 2xy j - 4xk, then evaluate jJJ V.JFdv where v bounded

V

by the planes Jt = 0,_y = 0,z = 0 and 2x + 2y + z = 4.

OR

Find the constants a,b,c so that

F (x + 2y + az)i + [bx-3y-z) j + (4x + cy + 2z)k is irrotational.

   x² 3v²

IfF = grad, show that = - + z² + 2xy + Axz - yz.

2 2

Verify Stokes theorem for F = (x² + y² j / 2xy j taken round the rectangle bounded by the lines x = 0_>(y = 0, = b. .    7*

Attachment: cochin-university-of-science-and-techology--cust--2007-b-tech-civil-engineering-36779-1.pdf

Earning:   Approval pending.